# Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup

From Groupprops

## Statement

Suppose is a finite group, is a subgroup, and is a field whose characteristic does not divide the order of (we do not require to be a splitting field, though the splitting field case is of particular interest).

Then, the Maximum degree of irreducible representation (?) of over is less than or equal to the product:

(maximum degree of irreducible representation of over ) times (index of in , i.e., )

## Facts used

## Related facts

## Proof

### Proof in characteristic zero

**Given**: is a finite group, is a subgroup, and is a field of characteristic zero. has index in . is the maximum of the degrees of irreducible representations of over .

**To prove**: has an irreducible representation over whose degree is at least .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let be the character of an irreducible representation of of degree . | is the maximum of degrees of irreducible representations of . | |||

2 | Let be the character of an irreducible representation of arising as a subrepresentation of . | is a subgroup of the finite group . | |||

3 | The inner product is a positive integer, because is the character of a subrepresentation of the representation affording . (Note: Over a splitting field, the inner product is precisely the multiplicity, but it may be bigger in general if is not absolutely irreducible). |
We need to have characteristic zero to make sense of positive integer. | |||

4 | Fact (2) | ||||

5 | is a positive integer, indicating that is an irreducible constituent of . | Steps (3), (4) | |||

6 | The degree of , which is , is less than or equal to the degree of | Steps (1), (5) | |||

7 | The degree of is times the degree of | has index in . | Definition of induced representation. | ||

8 | The degree of is at least | Steps (6), (7) |